Many subatomic particles, including certain atomic nuclei, are characterized by an angular momentum or spin. A spinning nucleus, commonly referred to simply as a "spin," is susceptible to disturbance when under the influence of fluctuations in the electromagnetic field. If a magnetic field H.sub.0 is imposed upon a "lattice" (a commonly accepted term for an aggregation of particles undergoing study), the spins, which constitute magnetic moments, tend to orient or align themselves with H.sub.0. Moreover, the spins precess about the direction of this field according to a phenomenon known as "Larmor Precession." Larmor precession occurs at a frequency-- "the Larmor frequency" --that is proportional to the magnitude of the applied field.
If field H.sub.0 is disturbed, the field alignment of the spins that constitute the lattice is also disturbed. This effect is particularly evident when field H.sub.0 is disturbed by an RF field bearing a relationship to the Larmor frequency of the lattice. As with systems generally, a lattice most easily absorbs energy at its own characteristic or resonant frequency, the Larmor frequency. The absorption of energy from an oscillating field H.sub.1 by a lattice that is already under a magnetic field H.sub.0 leads the spin axes to precess about the magnetic field resulting from the combination of H.sub.1 and H.sub.0. Such shifting or flipping is most efficiently accomplished by the application of a field H.sub.1 that is polarized in a plane orthogonal to H.sub.0 and which rotates at the Larmor frequency.
Observations of shifting or flipping of spins following the cessation of field H.sub.1 have revealed that the spins return to the ground state associated with H.sub.0 according to two first order relaxation processes. Prior to fully relaxing to their ground states, the spins produce electromagnetic signals having characteristics that are capable of detection, and from which inferences of enormous practical importance can be drawn. Such signals provide the basis for generating NMR images of human organs, for example.
The behavior of nuclear spins in magnetic fields as described above is illustrated in FIGS. 1-3. A lattice (not shown) may placed for study in a magnetic field H.sub.0 oriented in the z direction of a three dimensional reference frame (x, y, z). This and other magnetic fields necessary for conducting NMR analyses may be generated by a Philips Medical Systems (Best, Netherlands) Gyroscan NT.RTM., or other suitable device manufactured by various vendors, including the General Electric Company, Inc., and Siemens A.G. Under the influence of field H.sub.0, a spin, .mu., of the lattice precesses about H.sub.0 and the z axis. Absorption by a particular spin .mu. of energy from a rotating field H.sub.1 tips the spin away from H.sub.0 by an amount .theta., but it does not affect the rate of precession. This precession-rate occurs at the Larmor frequency, and is thus given by the Larmor equation: EQU .nu.=-.gamma.H.sub.o /2.pi.(Hz)
where .lambda. is the magnetogyric ratio, a constant for a particular spin. Clearly, it would be intractable to track the behavior of particular spins. Instead, as shown in FIG. 2, one can identify a macroscopic or net magnetization M, representing an aggregate magnetization for the spins (.mu..sub.0, .mu..sub.1, .mu..sub.3, .mu..sub.4, . . . .mu..sub.n) in the lattice of interest. The net magnetization M provides a convenient entity to which one can refer in respect of the behavior of the lattice as a whole. In the most simple case, when magnetic field H.sub.0 is applied in the z direction, magnetization M is also oriented in the z direction, about which the spins of the lattice precess. This case is illustrated in FIG. 2.
That the orientation of magnetization M and the axis of precession of the spins of a lattice vary according to fluctuations in the electromagnetic field implies that the axis of precession can be intentionally manipulated. More particularly, the application of an RF field H.sub.1 in a given direction imposes a torque along that direction and according to the well-known right hand rule. As shown in FIG. 3, a field H.sub.1 in the y direction imposes a torque on M that causes it to rotate away from z and about the y axis. The rate of this migration is linearly dependent upon the magnitude of the applied field. This behavior is predicted by the Bloch equation: EQU dM/dt=.gamma.M.times.B.
The RF field H.sub.1 may be circularly or linearly polarized in the x-y plane and rotated at the Larmor frequency in order to impart energy to the spins, and thereby rotate their axes of precession. The Larmor frequency, however, may be in the neighborhood of millions of cycles per second (Megahertz or Mhz). The resulting disturbance of the spins, which are initially spinning and precessing about z, is accordingly rather complicated and difficult to track. This difficulty can be overcome, however, by the adoption of a frame of reference (or, simply, a frame) that rotates about z in the same direction and at the same rate that the field H.sub.1 rotates. This coordinate system is commonly known as "the rotating frame" and is to be contrasted with a static (or laboratory) frame. The reader may analogize the rotating frame in the nuclear spin context to a rotating frame that every human being implicitly employs at every moment. A common example of a rotating frame continuously (although perhaps unwittingly) employed by everyone at every moment treats the surface of the earth and its objects as fixed, rather than rotating every 24 hours and translating with a velocity of over 1000 miles per hour. The perception of motion is suppressed by recognizing only a reference frame that rotates with respect to inertial space at a rate of once a day.
The motion of the magnetization vector M in a magnetic field H in the rotating frame is given by the following equation: EQU (dM/dt).sub.rot =.gamma.M.times.H-.omega..times.M
This equation can be rewritten as: EQU (dM/dt).sub.rot =.gamma.M.times.H.sub.eff
where EQU H.sub.eff =H+.omega./.gamma.
These equations state that the change in the magnetization with time as viewed within the rotating frame varies linearly with the product of the magnetization and an effective magnetic field H.sub.eff, and that this effective field H.sub.eff is the sum of the magnetic field H plus the rotational (angular) velocity divided by the magnetogyric ratio. The effective magnetic field H.sub.eff is simply equal to H.sub.1, if H.sub.1 and the rotating frame are at resonance with the Larmor frequency of the spins. The effective field H.sub.eff differs from H.sub.1 if it is "off-resonance." An off-resonance condition can result from practical limitations on the degree to which magnetic fields can be controlled and made uniform. This non-uniformity, and the off-resonance H.sub.eff to which it gives rise, represent the core of the problem solved by the present invention.
When a magnetization M and the associated spin precessions under a field H.sub.0 are exposed to a resonant RF field H.sub.1 in the direction of the positive x-axis, it begins to move or "flip" down toward the positive y-axis. Upon cessation of an applied field, the spins transfer energy to their surroundings, with respect to which they return to equilibrium, and turn back toward the z direction under H.sub.0. This first order relaxation process is characterized by two quantities: a time constant T.sub.1, known as the "longitudinal" or "spin-lattice relaxation time," in which spins transfer energy to the lattice; and a time constant T.sub.2, known as the "transverse" or "spin-spin relaxation time," according to which spins transfer energy to each other. When the relaxation of the magnetization M is along the magnetization, it is characterized by T.sub.1. If M relaxes when it is aligned with H.sub.eff in a rotating frame, it is characterized by the T.sub.1.rho. (.rho., the Greek letter r, signifying the rotating frame).
Although nuclear magnetic resonance (NMR) can be observed in several ways, one commonly used approach uses short bursts or pulses of RF power at a discrete frequency. Called, pulse methods or free precession techniques, they involve the observation of a nuclear spin system after the RF is shut off.
"Spin locking" is one useful technique for reducing effects of inhomogeneity in H.sub.0. In short, spin locking first applies a pulse to rotate M about a particular axis, say the y axis, by applying H.sub.1 in form of a 90.degree. pulse in the y direction. As shown in FIG. 3, M has rotated an angle .theta. about the y axis and towards the x axis. Once the 90.degree. pulse is completed, a 90.degree. phase change in H.sub.1 has been effected, so that H.sub.1 points along the x axis, collinear with M. With H.sub.1 lying along the same line as M, the former applies no torque on the latter and M is held in position or "locked" on the x axis. This locking process, and the knowledge of characteristic values for T.sub.1.rho. for various types of materials, permit T.sub.1.rho. to serve as the basis for certain imaging techniques and other analytical methods.
A conventional short hand notation summarizes the pulses in a sequence in the following form: EQU Ai-lockj-Ai,
where: (1) a first pulse causes a "flip down" rotation having an angular magnitude A about an axis i; (2) a locking pulse aligned along j, the axis of magnetization at the end of step 1, is applied to maintain the magnetization in that direction; and (3) a negative "flip back" rotation of magnitude A is applied about the i axis (negative direction or 180.degree. phase shift indicated by underscore). In the absence of off-resonance or other errors, the set of pulses in the above sequence should return the magnetization to alignment with an axis, along which it was aligned prior to the pulse sequence.
Known spin-locking and flip back techniques, however, tend to lock the magnetic field inefficiently and to place the magnetization M at a location that is imprecisely known. This imprecision arises from the existence of a deviation or off-resonant component, .epsilon., (also referred to here as resonance offset) between the direction of the effective magnetic field H.sub.eff and the intended direction of that field. This off-resonance component in turn results from spatial variations in the primary magnetic field H.sub.0. An undesired result of imprecision in directing the magnetization and spin lock inefficiency is that they can undermine imaging or other processes, the quality of which depends on the efficiency of the lock and the accuracy with which the direction of M is known.
An example of a known prior art spin lock technique is shown in FIGS. 4-6B and 13. (See Brief Description of the Drawings, below). FIG. 4 shows the pulse sequence 90y-lockx-90y. The horizontal axis in FIG. 4 measures time, while the vertical axis measures the amplitude of H.sub.1. In the illustrated sequence, the amplitudes of all illustrated pulses are arbitrarily chosen as unity (1).
One view of the effect on the magnetization of the pulse sequence illustrated in FIG. 4 is provided in FIG. 5 (macroscopic view) and in FIGS. 6A and 6B (microscopic view). The magnetization is assumed to be aligned along the z axis at t=0. On application of the 90.degree. "flip down" pulse, the magnetization is swept down to the x-y plane. As a result of an off-resonance term, .epsilon., however, the magnetization does not lie along the x axis (as desired), but is offset by one grid unit, defined for convenience to be equal to .epsilon.. (The value .epsilon. coincides with the spacing of the grids in all of FIGS. 5, 6A and 6B). The lock pulse is then applied in the x direction, along which axis the magnetization is believed to lie. In response to the lock pulse, the magnetization precesses about the x axis. The component of M perpendicular to x may be lost, M spiraling in to the x axis, in a time called T.sub.2.rho., which may be more or less than the duration of the desired spin lock. Any remaining perpendicular component will likely be a nuisance because its direction at the end of the lock is unknown. The magnetization arrives at a point in the x-z plane that is offset by one unit (.epsilon.) above the x axis. The magnetization is held at that position for the duration of the lock pulse, A 90.degree. flip back pulse is then applied in an attempt to restore the magnetization to the z axis position. Characterized by the same offset term, however, the flip back pulse does not return the magnetization to this direction, but leaves it in a direction in which it intersects the unit sphere at a point one unit (.epsilon.) to the right of the z axis and one unit past it, in the -x direction.
The efficiency of this known spin lock approach is shown in FIG. 13, which plots contours for the fraction of original magnetization remaining after a pulse (no relaxation) along the z axis as a function of the error in RF strength (defined as a fraction of the nominal strength) (vertical scale) and of the resonant frequency error (.epsilon., horizontal scale). In FIG. 13, the solid contours are as described. The dotted contours describe the case in which the 90.degree. pulses are much stronger than the lockx pulse in FIG. 4.
Although the significance of the plot of FIG. 13 will become more evident in the context of the discussion of the methods according to the present invention, a 95% efficiency region of the plot is rather tightly confined around the origin--where both types of error are at their lowest. Efficiency also falls off rather precipitately at higher error rates. The plot of FIG. 13 therefore indicates that spin lock efficiency using the above-described method is highly sensitive to both types of error. This sensitivity, particularly to resonant frequency error, gives rise to a variety of difficulties with NMR imaging and other applications, since spin lock inefficiency leads directly to loss of signal in whatever procedure is being performed.